Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Calculus in Motion: Kinematics

In one sentence. Displacement, velocity and acceleration as a chain of derivatives — the exam's favourite way to dress calculus in a story. Higher Tier.

Calculus — one of 2 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Galileo Galilei. Learn Calculus in Motion: Kinematics, teach it, then defend it against Galileo Galilei.

Where this sits in the specification

Specification 4MA1, 3.4 Calculus — kinematics (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 3.4 — Calculus — kinematics (Higher)

1 · Understand it

What it is

If s(t) is displacement (position) at time t, then velocity is v = ds/dt and acceleration is a = dv/dt: each is the derivative of the one before. 'At rest' means v = 0; 'maximum height' also means v = 0 (the moment of hanging still at the top).

Why it works

Velocity IS the rate at which position changes — that's its definition, and 'rate of change' is exactly what differentiation computes. Acceleration is the rate the velocity changes. The chain s → v → a is one idea applied twice.

When a question is asking for it

Any question giving displacement or height 'as a function of t': asked for velocity → differentiate once; acceleration → twice; 'when is it at rest / at its highest?' → set v = 0 and solve for t.

How to hold it in your head

Ride along with the particle. s says where you are, v says what the speedometer reads, a says how hard you're pressed into the seat. Differentiating moves you one instrument down the dashboard.

2 · The mistakes that cost marks

  • Setting s = 0 for 'at rest'. Rest is about MOTION stopping: v = 0. (s = 0 means 'back at the start' — different event.)
  • A negative velocity read as an error. It means moving BACKWARDS — direction, not mistake.
  • Zero acceleration read as 'stopped'. It means the velocity isn't changing — possibly cruising at high speed.
  • For maximum height, maximising h by trial instead of solving dh/dt = 0.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

s = t³ − 6t² + 9t (metres, t in seconds). Find the velocity at t = 2.

  1. v = ds/dt = 3t² − 12t + 9.
  2. At t = 2: 12 − 24 + 9 = −3 m/s.
  3. The minus means it's currently moving backwards.

Answer: −3 m/s

A ball's height is h = 20t − 5t². Find its maximum height.

  1. At the top, the ball hangs still: dh/dt = 20 − 10t = 0 → t = 2.
  2. Height then: h = 40 − 20 = 20 m.

Answer: 20 m

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — A particle's displacement is s = 5t² metres after t seconds. Work out its velocity at t = 3, in m/s. (answer in m/s)
  2. Grade 7 — s = t³ − 6t² + 9t. Work out the velocity at t = 2, in m/s. (answer in m/s)
  3. Grade 8 — For the same particle, s = t³ − 6t² + 9t, work out the FIRST time at which the particle is at rest.
  4. Grade 9 — Still with s = t³ − 6t² + 9t, work out the particle's acceleration at t = 2, in m/s². (answer in m/s²)
  5. 9+ — A ball is thrown upwards; its height is h = ut − 5t² metres after t seconds, where u is a positive constant. The maximum height the ball reaches is 45 m. Work out the value of u.

Ask the person who made it

Inside Incandio you can put a question to Isaac Newton, who answers knowing exactly which module you have just finished: “Position, velocity, acceleration — each is the rate of change of the one before, and your calculus connects them. Walk me through how differentiating turns 'where the apple is' into 'how fast it falls'.”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Calculus in Motion: Kinematics, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Calculus in Motion: Kinematics works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Calculus in Motion: Kinematics against Galileo Galilei — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 9 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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